For \(\rho =0\) , the system (REF ) becomes the Hunter-Saxton (HS) equation [1]},
which models the propagation of weakly nonlinear orientation waves in a massive
nematic liquid crystal director field.
Here, \(u(x,t)\)  stands for the director field of a nematic liquid crystal,
\(x\)  is a
space variable in a reference frame moving with the linearized wave velocity,
and \(t\)  is a slow time variable.
The field of unit vectors \(( \cos {u(x,t)}, \sin {u(x,t)})\)  describes
the orientation of the molecules [2]}, [1]}, [4]}.
